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FORCED VIBRATION TESTING AND ANALYSIS OF PRE- AND POST-
RETROFIT BUILDINGS
A Thesis
Presented to
the Faculty of California Polytechnic State University
San Luis Obispo, California
In partial fulfillment
of the Requirements for the Degree
Master of Science in Architecture with a Specialization in Architectural Engineering
by
Erica Dawn Jacobsen
June 2011
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© 2011
Erica Dawn Jacobsen
ALL RIGHTS RESERVED
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COMMITTEE MEMBERSHIP
TITLE: Forced Vibration Testing and Analysis of Pre- and Post-
Retrofit Buildings
AUTHOR: Erica Dawn Jacobsen
DATE SUBMITTED: June, 2011
COMMITTEE CHAIR: Graham Archer, Ph.D., P.Eng.
COMMITTEE MEMBER: Cole McDaniel, Ph.D., P.E.
COMMITTEE MEMBER: Kevin Dong, S.E.
COMMITTEE MEMBER: Deborah Wilhelm, M.A., M.P.S.
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ABSTRACT
Forced Vibration Testing and Analysis of Pre- and Post- Retrofit Buildings
Erica Dawn Jacobsen
The primary goal of the thesis was to detect the retrofit through vibration testing
of both buildings. The secondary goal focused on correctly identifying the behavior of
the building through FVT, comparing that behavior to computational model predictions,
and determining the necessary level of detail to include in the computational modeling.
Forced vibration testing (FVT) of two stiff-wall/flexible-diaphragm buildings yielded
natural frequencies and mode shapes for the two buildings. The buildings were nearly
identical with the exception that one had been retrofitted. Both buildings were comprised
of concrete shearwalls and steel moment frames in the north/south direction and moment
frames in the east/west direction. The retrofit strengthened the moment connections and
added braces to the perimeter walls in the east/west direction.
The natural frequencies were found through FVT by setting a 30-lb shaker on the
roof of both buildings and sweeping through a range of frequencies in both the east/west
and north/south directions. Accelerometers were placed on the building to detect the
accelerations. The peaks on the Fast Fourier Transform (FFT) graphs indicated the
frequencies at which the structure resonated. Mode shapes were tested for by placing the
shaker in a position ideal for exciting the mode and setting the shaker to the natural
frequency detected from the FFT graphs. The accelerometers were placed around the
roof of the building to record the mode shape.
After testing, computational models were created to determine if the models could
accurately predict the frequencies and mode shapes of the buildings as well as the effect
of the retrofit. A series of increasingly complex computational models, ranging from
hand calculations to 3D models, were created to determine the level of detail necessary to
predict the building behavior. Natural frequencies were the primary criteria used to
determine whether the model accurately predicted the building behavior. The mid-
diaphragm deflection and base shear from spectral analysis were the final criteria used to
compare these select models.
It was determined that in order to properly capture the modal behavior of the
building, the sawtooth framing, major beams, and the lateral-force-resisting-system
(LFRS) must be modeled. Though the mode shape of the building is dominated by the
flexible diaphragm, the LFRS is necessary to model to accurately predict both the natural
frequency of the building as well as the diaphragm deflection.
Keywords: Ambient Vibration Testing, Forced Vibration Testing, Retrofit, Flexible
Diaphragm, Mode Shapes, Natural Frequencies
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TABLE OF CONTENTS
TABLE OF CONTENTS .............................................................................................................................. v
LIST OF TABLES .......................................................................................................................................vi
TABLE OF FIGURES ............................................................................................................................... vii
1.0 INTRODUCTION ............................................................................................................................... 1
1.1 OBJECTIVE ......................................................................................................................................... 1 1.2 BUILDING DESCRIPTION..................................................................................................................... 2 1.3 BACKGROUND ................................................................................................................................... 5
2.0 LITERATURE REVIEW ................................................................................................................... 7
2.1 PREVIOUS TESTING METHODS ........................................................................................................... 7 2.2 STIFF STRUCTURES WITH FLEXIBLE DIAPHRAGMS ............................................................................ 8
3.0 VIBRATION TESTING FOR MODAL CHARACTERISTICS .................................................. 12
3.1 EQUIPMENT ...................................................................................................................................... 12 3.2 TESTING FOR NATURAL FREQUENCIES ............................................................................................ 12 3.3 TESTING FOR MODE SHAPES ............................................................................................................ 22 3.4 DAMPING RATIOS ............................................................................................................................ 35 3.5 SUMMARY........................................................................................................................................ 39
4.0 COMPUTATIONAL MODELING ................................................................................................. 41
4.1 HAND CALCULATIONS ..................................................................................................................... 41 4.2 COMPUTATIONAL MODELING .......................................................................................................... 46 4.3 MODAL ANALYSIS ........................................................................................................................... 63 4.4 SUMMARY........................................................................................................................................ 70
5.0 BUILDING RESPONSE ................................................................................................................... 73
6.0 CONCLUSIONS ................................................................................................................................ 79
7.0 WORKS REFERENCED LIST ........................................................................................................ 84
8.0 WORKS CONSULTED LIST .......................................................................................................... 86
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LIST OF TABLES
Table 1: Hand Calculation Results for NS Direction........................................................ 43
Table 2: Hand Calculation Results for EW Direction ...................................................... 45
Table 3: Membrane Model Results for the NS Direction ................................................. 47
Table 4: Diaphragm Model With Sawtooth Results ......................................................... 51
Table 5: 3-D Model Results for NS Direction .................................................................. 54
Table 6: 3-D Model Results for EW Direction. ................................................................ 61
Table 7: Comparison of Responses for Different Unretrofit Building Models ................ 75
Table 8: Comparison of Retrofit Building Responses ...................................................... 76
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TABLE OF FIGURES
Figure A: Unretrofit Building Framing Plan and Section ................................................... 3
Figure B: Retofit Building Framing Plan............................................................................ 4
Figure C: Accelerometer Layout for NS Testing .............................................................. 13
Figure D: Natural Frequency Sweep in NS Direction of Unretrofit Building .................. 15
Figure E: Natural Frequency Sweep in EW Direction of Unretrofit Building ................. 16
Figure F. Accelerometer Layout for EW Testing ............................................................. 17
Figure G: Enlarged View of EW Sweep of Unretrofit Building ...................................... 18
Figure H: Natural Frequency Sweep in NS Direction of Retrofit Building ..................... 19
Figure I: Natural Frequency Sweep in EW Direction of Retrofit Building ...................... 20
Figure J: Enlarged View of EW Sweep of Retrofit Building ........................................... 21
Figure K: Mode 1 Equipment Setup and Expected Mode Shape (Plan View) ................. 22
Figure L: Mode 2 Equipment Setup and Expected Mode Shape (Plan View) ................. 24
Figures M: Unswept Mode 1of the Unretrofit Building ................................................... 25
Figures N: Unswept Mode 2 of the Unretrofit Building ................................................... 25
Figure O: Pure Modes 1, 2, and 3 Used in Theoretical Example ..................................... 28
Figure P: Polluted Mode 2 Used in Theoretical Example ................................................ 28
Figure Q: Swept Mode 2 of Unretrofit Building .............................................................. 30
Figure R: Cross Section of Existing Roof......................................................................... 32
Figure S: Unswept Mode 1 of the Retrofit Building......................................................... 33
Figure T: Unswept Mode 2 of the Retrofit Building ........................................................ 33
Figure U. Swept Mode 2 for the Retrofit Building ........................................................... 34
Figure V: Damping Ratios for the Unretrofit Building .................................................... 36
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Figure W: Damping Ratios for the Retrofit Building ....................................................... 38
Figure X: Computer Model With Vertical Members Included ......................................... 53
Figure Y: Section Showing Moment Frame Beams at Different Elevations .................... 56
Figure Z: Ambient Vibration of the Unretrofit Building in the EW Direction ................. 66
Figure AA: Ambient Vibration of the Unretrofit Building in the NS Direction .............. 67
Figure BB: Ambient Vibration of the Retrofit Building in the EW Direction ................. 68
Figure CC: Ambient Vibration of the Retrofit Building in the NS Direction .................. 69
Figure DD: Response Spectra Used in Comparison of Models ........................................ 73
1.0 Introduction 1
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
1.0 INTRODUCTION
This thesis explores the behavior of two nearly identical concrete-shearwall
buildings with flexible diaphragms, one of which has been seismically retrofitted. The
purpose of this thesis is to use forced vibration testing (FVT) to detect the retrofit in the
buildings and compare those test results to computational model predictions. The modal
behavior of the two buildings is used as criterion for determining whether a
computational model can accurately predict the building behavior and more specifically
the retrofit. Mode shapes and natural frequencies are compared to all the computational
models; select models are also analyzed by using spectral analysis. From this analysis
the diaphragm deflection, base shear, and wall base shear are compared between the
models; this comparison is used as an additional criteria for judging to what extent the
models differ from one another. Through that comparison it is determined what detail is
necessary in a model to accurately predict the behavior of the buildings.
1.1 Objective
The objective of this thesis is to use FVT to determine what effect the retrofit
has on the building dynamics. The results from the testing are then compared to
computational model predictions to conclude whether the computational model correctly
predicts the effect the retrofit has on the building. Through the testing, modal
information is gathered that is then compared to computational model predictions.
Currently, there is little research regarding the modal behavior of buildings with
stiff shearwalls and flexible diaphragms. Another objective of this thesis is to analyze the
1.0 Introduction 2
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
ability of different computational models to effectively predict the natural periods as well
as the mode shapes identified from the FVT. The level of detail necessary to effectively
model the building is determined through the process of comparing computational results
with FVT results. The criterion for determining the accuracy of the computational model
is based on the natural frequencies and mode shapes.
1.2 Building Description
The experimentation for this thesis is performed on two of six nearly identical
buildings, one of which was retrofitted in 2005 and one that was retrofitted after the
testing in 2010. For the purpose of this thesis, the buildings will be referred to as the
retrofit building (RB) and the unretrofit building (URB). In order to capture the pre-
retrofit and post-retrofit behavior of a building, it would typically be necessary to wait for
construction of the retrofit to finish, therefore delaying the testing. By testing two nearly
identical buildings, one with a retrofit and one without, waiting for construction was
unnecessary, making these two buildings ideal for testing.
The buildings are located in Northern California and are both 120,000 square
feet, 2-story commercial buildings with a waffle slab at the second floor and light frame
wood and steel roof. At the bottom story there are three shearwalls around the perimeter
and concrete columns on the interior. At the top story there are steel moment frames in
the east/west direction and concrete shearwalls and intermediate moment frames in the
north/south direction. Since the bottom story has concrete shearwalls on three sides and
is partially underground due to the sloped site, the bottom story is over ten times stiffer
1.0 Introduction 3
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
than the top story, so according to the International Building Code (IBC 2006), the top
story can be analyzed as a single-story building. The framing plan and cross section of
the URB, as used in the analysis, is shown in Figure A, below.
Figure A: Unretrofit Building Framing Plan and Section
Source: Jacobsen (2011)
The structure for both buildings has a hybrid roofing system consisting of steel
girders and beams with wood joists and a plywood roof. A large percentage of the roof is
also composed of sawtooth framing, as seen in Figure A. The plywood diaphragm is
made more flexible by the sawtooth and because of the flexible diaphragm, each mode of
both buildings only moves in one direction, east/west (EW) or north/south (NS), and is
unaffected by modes in the other direction.
1.0 Introduction 4
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
The difference between the two buildings is the additional braces at the second
story in the EW direction of the RB. The braces were added to stiffen the building and to
reduce the load in the existing moment frames. The first story of the RB was determined
to have adequate strength, so it was not retrofit. The framing plan for the RB is shown
below in Figure B.
Figure B: Retofit Building Framing Plan
Source: Jacobsen (2011)
As illustrated in Figure B, the only significant change to the building is the
addition of braces in the (EW) direction. This retrofit scheme would cause a potential
change in the EW mode of the building, as well as, increase in the natural frequencies.
Because the retrofit was primarily in the EW direction, the natural frequencies and mode
shapes in the NS direction should not have been affected. This is proven in the testing
and will be discussed in Section 3.
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Forced Vibration Testing of Pre- and Post- Retrofit Buildings
1.3 Background
The first step in most analyses is to determine the building’s natural periods as
well as the first several mode shapes and damping ratios. Computational models of
existing buildings can be created that will output these mentioned responses.
Experimental testing of the building can be employed as well. Ambient vibration testing
(AVT) and forced vibration testing (FVT) have been accepted methods for finding the
fundamental periods of a building since the 1930s (Ivanovic 2000).
AVT requires placing accelerometers along the building and measuring the
vibrations of the building caused by everyday occurrences such as mechanical equipment,
wind, and traffic. In contrast, FVT involves shaking the building, typically the roof. The
responses of the building to this force are then measured by accelerometers. The field
effort involved in AVT is less than that for FVT because less equipment and set-up is
involved (Trifunac 1972). However, the smaller the vibrations, the more difficult it is to
detect the building’s responses. Unless the frequency of those vibrations resonates with
the building, it may be difficult to detect the natural frequency of the building with AVT.
For this reason, FVT may be preferred regardless of the additional setup and equipment
required.
Most FVT tests are preformed on tall, rigid diaphragm buildings. It should be
noted that stiff buildings with flexible diaphragms are an often-overlooked category of
seismic analysis, made apparent in the literature review. This oversight is also evident in
computer programs. Many computer programs are designed for rigid diaphragms so their
ability to predict the behavior of a flexible diaphragm is lacking. ETABS, a common
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Forced Vibration Testing of Pre- and Post- Retrofit Buildings
engineering modeling program, restricts the option to rigid or semi-rigid diaphragm. In
addition, the program does not include plywood or wood on the list of pre-programmed
materials. Although these obstacles can be overcome in ETABs, caution should be
exercised when trying to model flexible plywood diaphragms.
Some engineers believe that the natural frequencies of the building are determined
by the stiffness of the lateral-force-resisting-system (LFRS), such as braced frames,
moment frames, or shear walls. However, recent research, discussed in the literature
reviews, reveals that in a building with significantly stiffer walls than diaphragm, the first
several natural frequencies are governed by the flexible diaphragm. Consequently, the
mode shapes of the building resemble those of a flexible beam rather than the
conventional rigid-body translational mode shapes. FEMA 310 (FEMA 1998) has
provisions regarding the retrofit of such buildings, but these provisions depend upon the
natural frequency of the building. It is only in recent testing that researchers are
beginning to understand the flexible diaphragm behavior in stiff buildings, and more
testing is required before an accurate prediction of the natural frequencies is codified,
which is also discussed in the literature review.
2.0 Literature Review 7
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2.0 LITERATURE REVIEW
The buildings’ behavior is best represented by the behavior of a stiff building with
a flexible diaphragm similar to masonry buildings with plywood roofs. Research
regarding the behavior of these buildings, as well as the testing methods, is explained
below.
2.1 Previous Testing Methods
Ambient and forced vibration testing has been explored both separately and in
comparison to each other. Ivanovic (2000) performed an ambient vibration test on a
building in Van Nuys, California and Luco, Trifunac, and Wong (1988) performed a
forced vibration test on the Milikan Library in Pasadena, California. The definition used
in this thesis for forced and ambient vibration arose from these sources. Trifunac (1972)
tested two buildings, a 22-story steel frame building and a nine-story concrete frame
building, and determined that forced and ambient vibration yielded similar results.
However, both buildings discussed by Trifunac were considerably taller than the
buildings investigated in this thesis and, therefore, more susceptible to ambient
vibrations. Because there were fewer case studies about vibration testing of shorter
buildings, and fewer still regarding buildings with flexible diaphragms, both FVT and
AVT were performed despite Trifunac’s findings. However, Trifunac’s conclusion that
forced and ambient vibration yield similar results eventually proved to be true for the
testing done in this thesis as well.
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McDaniel and Archer (2009) detail the AVT and FVT set-up as well as the
equipment used in this thesis. In their conference paper, McDaniel and Archer used FVT
to identify the first three natural frequencies and mode shapes of a two-story concrete
building. In their paper they proved that a 30-lb shaker could excite the natural
frequencies of a full-size building, results that were later corroborated with computer
models. Their research provided the basis for the experiment done in this thesis.
2.2 Stiff Structures with Flexible Diaphragms
There is a movement in the engineering community to correct the code provisions
for stiff structures with flexible diaphragms. The SEAOC Seismology Committee (2008)
propose that an entirely new approach in determining the natural frequency of stiff
structures with flexible diaphragms is needed. The current lumped-mass model only takes
into account the rigidity of the LFRS and not the diaphragm. Though research has
investigated new methods for determining these buildings’ natural frequencies, more
research is required before code changes can occur.
Freeman, Searer, and Gilmartin (2002) proposed that code provisions do not
accurately describe the behavior of these structures when excited by earthquake ground
motions. Their article suggests a method for finding the earthquake loads in a building
based on the first two modes of motion; the first fundamental mode is the in-plane
bending of the diaphragm as though it were a horizontally supported beam and the second
fundamental mode of vibration is due to the out-of-plane bending of the pre-cast concrete
tilt-up walls. While the first mode was found in the testing for this thesis, the second
2.0 Literature Review 9
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mode was not present because there were no shearwalls along the longitudinal edges of
the buildings.
Studies suggest that a flexible diaphragm behaves similar to that of a simply
supported beam. Rao (2004) presents equations for finding the natural frequencies of a
simply supported beam by looking at the rotary inertia of the beam alone, the shear
stiffness alone, and then with neither parameters. The analysis based on the equations by
Rao (2004) was used as a preliminary method to compute the natural frequencies of the
buildings, but ultimately the model was too simplistic to capture the modal behavior of
the buildings.
There was some debate on whether the flexible diaphragm remained linear
through high excitation. The accelerations exerted on the buildings in this thesis are low
amplitude, not enough to overcome the internal frictions of connections, so the
fundamental periods should be lower than those found if the buildings were excited at
earthquake-level forces. Paquette and Bruneau (2005) performed a full scale, shake table
test on a one-story masonry building with a flexible diaphragm, and discovered that the
diaphragm in their experiment remained linear elastic through their pseudo-dynamic
testing. This conclusion suggests that the natural frequency of the diaphragm will not
shift at higher force levels.
In direct contrast to Paquette and Bruneau (2005), Rogers and Tremblay (2005)
believed the behavior of a flexible diaphragm was non-linear due to connection failure at
the panels. Camelo (2003) performed vibration testing on five wood buildings and also
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found that the natural frequency shifted as much as 40% as the amplitude of the
accelerations increased. The American Plywood Association (APA) deflection equation
(SEAONC 2001) for flexible diaphragms includes four components: bending
deformation, shear deformation, nail slip, and chord slip. Nail slip in particular does not
have a linear force-displacement relation and so the APA equation suggests that the
stiffness of the diaphragm is load dependant. In Freeman, Searer, and Gilmartin (2002),
they separated the period of the diaphragm into three categories: “(a) initial period at low
amplitude motion, (b) design level at moderate amplitudes below yield level and (c)
inelastic response at reasonably acceptable ductility limits.”
Based on this research, it seems probable that the period of the diaphragm shifts
when the accelerations have higher amplitudes, but due to the limitations of the testing
equipment, 30-lbs as the maximum force of the shaker, testing the two buildings at higher
amplitudes was not possible. There was also concern that a higher amplitude shaker
might have proven destructive to the building. While Freeman, Searer, and Gilmartin
(2002) don’t explain what levels of force they considered to be low amplitude, they do
state that level (b) was considered appropriate for design forces and level (c) should be
considered for maximum displacements. The 30-lb shaker used in the testing for this
thesis is well below design forces, and so the building frequencies found during testing
are at the lowest level, level (a). As a result of Camelo’s (2003) research it was
speculated that the frequencies found in the testing for this thesis are approximately 70%
2.0 Literature Review 11
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
of what they will be during an earthquake, but more research will have to be conducted to
state definitively.
3.0 Vibration Testing 12
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
3.0 VIBRATION TESTING FOR MODAL CHARACTERISTICS
Ambient and forced vibration tests were performed on both buildings to identify
the natural frequencies as well and the mode shapes of each building. The initial tests
swept for the natural frequencies and subsequent tests yielded the mode shapes.
3.1 Equipment
The FVT was performed with a linear shaker capable of delivering a 30-lb force.
A signal generator was used to set the shaker to vibrate at any frequency within 2Hz to
20Hz. The signal generator also had the option to sweep through a range of frequencies
in a two minute cycle. This function was used to locate the natural frequencies of the
building. The vibrations that the building exhibited were detected through the use of
piezoelectric 5-g accelerometers that were sensitive to a micro-g. Three accelerometers
were used to capture the NS translational, the EW translational, and the rotational
component of the accelerations.
3.2 Testing for Natural Frequencies
The accelerometers were first placed on the roof of the RB near the center of the
diaphragm facing the north direction. Accelerometers A and C were located in the
middle of the south edge of the diaphragm with A facing in the north direction and C in
the east. Accelerometer B was set facing the north direction 16' east of accelerometers A
and C as shown in Figure C, on the following page.
3.0 Vibration Testing 13
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure C: Accelerometer Layout for NS Testing
Source: Jacobsen (2011)
The difference between the A and B accelerometers was converted into a
rotational component by using the equation
Rot ′ "/′
. Eq. 1
where Rot is the rotational component (unitless),
A is the acceleration in accelerometer A (µgs),
B is the acceleration in accelerometer B (µgs),
and g is the acceleration due to gravity (in-s2/ft).
Though B was not required to be located sixteen feet from A, by placing the
accelerometers at that distance, the rotation of the building was approximately the same
amplitude as the translational accelerations, allowing all three components to be
displayed on the same plot.
3.0 Vibration Testing 14
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
An ambient vibration test was performed over a range of frequencies from 2 Hz to
8 Hz, shown in Figures D and E on the following pages. Ten two-minute recordings
were taken and then averaged to produce the ambient vibration graphs.
After the ambient vibration test was complete, the shaker was placed at the middle
of the south edge of the diaphragm facing NS direction to excite the first mode. The first
mode shape of the building is the diaphragm flexing between the rigid shear walls like a
beam. As stated previously, the NS and EW modes are uncoupled due to the flexibility
of the diaphragm. A flexible diaphragm has little rotational component in the first
several modes and mode shapes of a flexible diaphragm have motion in one direction
only. This modal behavior is because the diaphragm does not couple the NS and EW
motions into a single mode. If a diaphragm were completely flexible, then all mode
shapes would only have motion in a single direction. Therefore the EW and NS modes
are unaffected by each other.
The graph in Figure D on the following page shows the responses acquired at
different testing positions. The shaker faced the NS direction to isolate the NS mode,
faced the EW direction to isolate the EW mode, and then was turned 45º from the
orthogonal axes of the building, so all modes had the potential to be excited.
3.0 Vibration Testing 15
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure D: Natural Frequency Sweep in NS Direction of Unretrofit Building
Source: Jacobsen (2011)
Figure D shows the frequency range that was swept, and the peaks on the graph
indicate where the natural frequencies lie. As illustrated in Figure D, the natural
frequencies for the URB were 3.35 Hz and 6.18 Hz, for Mode 1 and Mode 2 respectively.
The peaks for AVT and FVT did line up, although the magnitude of the peaks is greater
with the FVT. Similarly, the FVT performed at an angle, position 2, has a smaller
amplitude than the FVT oriented directly NS.
The sweep in the EW direction, from 2 Hz to 10 Hz, for the URB is shown in
Figure E on the following page, and as in Figure D, the position of the shaker is shown in
the upper right corner of the figure.
3.0 Vibration Testing 16
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure E: Natural Frequency Sweep in EW Direction of Unretrofit Building
Source: Jacobsen (2011)
While the accelerometers for the NS direction were always located at the center of
the south side of the diaphragm, the EW direction had little response at the south edge of
the diaphragm, positions 2 and 3, so the accelerometers and shaker were relocated to the
east side of the building at position 1, shown on the following page in Figure F.
3.0 Vibration Testing 17
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure F. Accelerometer Layout for EW Testing
Source: Jacobsen (2011)
The expected mode shape of a flexible diaphragm, according to Freeman, Searer,
and Gilmartin (2002), is that of a simply supported beam, so by placing the shaker in the
middle of the east edge, the natural frequency should have been more easily excited than
when the shaker was placed on the edge of the diaphragm.
Though there is a large peak shown at 9 Hz in Figure E, a peak that sharp
indicates the oscillation of mechanical equipment, such as an HVAC unit, rather than a
building response. The width of the peak indicates the damping (Chopra 2007), therefore
a sharp peak indicates low damping which is inherent in mechanical equipment. On both
buildings there were HVAC units that did not appear to be isolated from the roof. The
sharp peak shown in Figure E is the natural frequency of the fan in one of the HVAC
units, not the building. Figure G on the following page shows an enlarged view of the
EW sweep.
3.0 Vibration Testing 18
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure G: Enlarged View of EW Sweep of Unretrofit Building
Source: Jacobsen (2011)
As seen in Figure G, the only peak appears to be at 7.1 Hz; however, the low
accelerations suggest that the peak is not an EW mode. Referring again to Figure D,
there was a peak in the NS direction at 7.1 Hz with a stronger response than the EW
direction; 50 µgs were recorded in the NS direction as opposed to 5 µgs in the EW. This
data suggests that the peak at 7.1 Hz was more likely a shadow of one of the NS modes,
rather than its own mode. Although testing was done in the EW direction multiple times
and in multiple positions, no natural frequencies between 2 Hz and 8 Hz were detected.
An AVT and a FVT were run in both directions for the RB as well. The AVT
detected accelerations in the range of 2 Hz to 8 Hz at the midpoint of the diaphragm as
seen in Figure H on the following page. After the ambient test, a forced vibration test
3.0 Vibration Testing 19
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
was run in the NS direction from 2 Hz to 8 Hz, and an angled sweep was performed from
2 Hz to 8 Hz. The position of the shaker is displayed on the diagram at the upper right
corner of the figure.
Figure H: Natural Frequency Sweep in NS Direction of Retrofit Building
Source: Jacobsen (2011)
The natural frequencies in the NS direction are clearly delineated in Figure H at
2.63 Hz and 5.58 Hz. Using the same method used with the URB, the accelerometers
were placed at the middle of the diaphragm.
The shaker cycles through all the frequencies in the range during the two minute
recording and only hits the natural frequency of the building for a moment before moving
onto the next frequency. The accelerometers are able to pick up that momentary
resonance before the shaker changes frequency, but there was concern that the natural
frequency might not be detected if the shaker was at that frequency for too brief time.
3.0 Vibration Testing 20
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Considering this concern, for the EW mode, the shaker and accelerometers were placed
on the east edge and the shaker swept from 2 Hz to 18 Hz in two different sweeps. The
first sweep was from 2 Hz to 9 Hz, and the second sweep was from 8 Hz to 16 Hz.
Similar to the unretrofit building, the EW sweep on the retrofit building did not
detect any natural frequencies between 2 Hz and 10 Hz. Figure I below shows the sweeps
and the areas where no clear peaks are apparent.
Figure I: Natural Frequency Sweep in EW Direction of Retrofit Building
Source: Jacobsen (2011)
The shaker and accelerometers were placed at different points on the buildings,
and the only apparent peak was in position 1 at 11 Hz, however the peak is beyond the
range of interest from a structural dynamics perspective. In addition, computational
models suggest that the fundamental frequency of the retrofit building should have been
3.0 Vibration Testing 21
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
anywhere from 2.5 Hz to 4 Hz but a closer view of the EW sweep, shown in Figure J
below, reveals no peaks.
Figure J: Enlarged View of EW Sweep of Retrofit Building
Source: Jacobsen (2011)
As illustrated in Figure J there are no peaks to indicate that the fundamental
frequency in the EW direction was in the range of 2 Hz to 6 Hz. In the NS readings the
various forced vibration tests as well as the ambient test lined up, although with differing
amplitudes. The readings for the EW direction did not line up, but since all the readings
were below 10 µgs the ambient vibrations could have distorted the readings differently
each time. The peak of the natural frequency should have been clear above the ambient
noise; that there is no peak in Figure J suggests there was no natural frequency in the
tested range.
-1.00E-06
1.00E-06
3.00E-06
5.00E-06
7.00E-06
9.00E-06
1.10E-05
1.30E-05
1.50E-05
2 3 4 5 6
Acc
eler
ati
on
(g
)
Frequency (Hz)
E/W Vibration Test of Retrofit Building
Forced Vibration (1)
Forced Vibration (2)
Forced Vibration (3)
Ambient Vibration (4)
3.0 Vibration Testing 22
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
3.3 Testing for Mode Shapes
After the natural frequencies for both buildings were determined, another round of
testing began to map the mode shapes, the natural shape of the building when vibrating at
a resonant frequency. The mode shapes were mapped for the first two natural
frequencies of each building in the NS direction. As seen in Figure G on page 18 and
Figure I on page 21, the natural frequencies in the EW direction were undetectable and so
the focus of the testing became centered on the NS direction of the buildings. In order to
map the mode shapes of both buildings, the shaker was placed at a point along the
diaphragm that would maximize one mode shape while minimizing the other mode
shapes. Figure K below shows the shaker positioned for Mode 1 and the expected mode
shape.
Figure K: Mode 1 Equipment Setup and Expected Mode Shape (Plan View)
Source: Jacobsen (2011)
3.0 Vibration Testing 23
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
For the first mode, the shaker was placed at the center of the diaphragm to
maximize the simple bending of the diaphragm. The center of the diaphragm was also a
node point for the second mode, assumed to be double bending, and so would minimize
the participation of Mode 2.
Accelerometers were placed at the middle and quarter points of the diaphragm, as
seen in Figure L on the following page, to record the behavior of the building.
Accelerometers were also placed at the edge of the building to determine whether the
accelerations at the concrete shear walls were zero. Verifying that the accelerations were
approximately zero confirmed that the diaphragm was accommodating the majority of the
deformation, just as the Freeman, Searer, and Gilmartin (2002) suggested.
Based on the sweep, the first natural frequencies of the RB and URB were 3.35
Hz and 2.63 Hz respectively. For the mode shape testing, the shaker was set to the first
natural frequency, and accelerometers were placed at the east quarter point while five
recordings were taken. To map the shape of the mode, tt was important to determine
whether the various points along the diaphragm were in phase with each other. To check
the phase of the middle point, accelerometers A & C were left at the quarter point and
accelerometer B was moved to the middle point. Once in place, the phase of the three
accelerometers was compared to determine whether the locations moved together or
opposite from each other. This “leap-frog” method was always used before moving the
accelerometers to the next point and when determining whether the north side and south
side of the diaphragm acted with or without each other. With most buildings the north
3.0 Vibration Testing 24
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
and south side would be assumed to act in unison, however with the large sawtooth, the
diaphragm could have been considered largely discontinuous, so it was possible that the
two sides would not act as one solid diaphragm. The “leap-frog” method of testing was
confirmed that the north and south sides did act in phase for the first mode.
For the second mode, the shaker was placed at quarter points: the west quarter
point for the RB and the east quarter point for the URB, to excite double bending.
Accelerations were taken at the same points as the first mode. Figure L below shows the
equipment placement as well as the expected shape.
Figure L: Mode 2 Equipment Setup and Expected Mode Shape (Plan View)
Source: Jacobsen (2011)
Once in place, the shaker was set to the second natural frequencies found from the
sweep, 5.58 Hz for the RB and 6.18 Hz for the URB. The accelerations were recorded in
the same manner as Mode 1, with at least five accelerations recorded at each po
with special attention to the phase angle of each location.
The mode shapes of the
diaphragm. The first two unswept modes of the unretrofit building are shown in Figure
M and N below.
Figures M & N: Unswept Mode 1 (Left) and Mode 2 (right) of the Unretrofit
Figures M and N plot roof displacement in relation to the position along the
diaphragm to show the mode shape found during testing. The double arrow in the figures
indicates where the shaker was placed to excite the mode shape
represent the shape of the north and south side of the diaphragm. The modes shown in
Figures M and N are orthonormalized, but unswept, so while Mode 1 appears in its final
form, Mode 2 is hardly discernible.
In Figure M, it is apparent that the north and south sides of the buildi
moving together in a single
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 100
Dis
pla
cem
en
t (f
t)
Length Along the Diaphragm
Unretrofit Building: Unswept Mode 1
3.0 Vibration Testing
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
the same manner as Mode 1, with at least five accelerations recorded at each po
with special attention to the phase angle of each location.
mode shapes of the RB and URB resemble a beam due to the flexible wood
diaphragm. The first two unswept modes of the unretrofit building are shown in Figure
: Unswept Mode 1 (Left) and Mode 2 (right) of the Unretrofit
Building
Source: Jacobsen (2011)
plot roof displacement in relation to the position along the
diaphragm to show the mode shape found during testing. The double arrow in the figures
indicates where the shaker was placed to excite the mode shape, and the two lines
e north and south side of the diaphragm. The modes shown in
are orthonormalized, but unswept, so while Mode 1 appears in its final
form, Mode 2 is hardly discernible.
, it is apparent that the north and south sides of the buildi
moving together in a single-bending shape with the maximum displacement in the
200
Length Along the Diaphragm
Unretrofit Building: Unswept Mode 1
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 100 200
Length Along the Diaphragm
Unretrofit Building: Unswept Mode 2
3.0 Vibration Testing 25
Retrofit Buildings
the same manner as Mode 1, with at least five accelerations recorded at each point and
resemble a beam due to the flexible wood
diaphragm. The first two unswept modes of the unretrofit building are shown in Figures
: Unswept Mode 1 (Left) and Mode 2 (right) of the Unretrofit
plot roof displacement in relation to the position along the
diaphragm to show the mode shape found during testing. The double arrow in the figures
and the two lines
e north and south side of the diaphragm. The modes shown in
are orthonormalized, but unswept, so while Mode 1 appears in its final
, it is apparent that the north and south sides of the building were
bending shape with the maximum displacement in the
200
Length Along the Diaphragm
Unretrofit Building: Unswept Mode 2
South Side
Experimental
North Side
Experimental
3.0 Vibration Testing 26
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
middle. It should be noted that in both Figures M and N only the five points with
diamond markers were recorded points. The line that connects the points is a spline to
connect the points, so aside from those five points, the mode shape is assumed.
Mode 2 was polluted with other modes, and it was necessary to sweep Mode 1 out
of it. In this thesis, it was assumed that the first mode was pure and so could be swept out
of mode 2. In Figure M there was a clear asymmetry in the mode shape, with the right
quarter point higher than the left quarter point. If Mode 1 were truly a pure mode it
should be symmetrical about the center of the diaphragm. This asymmetry suggests that
other modes were participating in Mode 1 which made sweeping Mode 1 out of Mode 2
less successful than had Mode 1 been pure.
Once the mode shapes were mapped the data was processed to extract the pure
mode. Due to modal participation from other modes, the raw data gathered was polluted,
and the other modes were swept using the Gram-Schmidt Orthogonalization equation
1
11
2122 ' Φ
ΦΦ
ΦΦ−Φ=Φ
M
MT
T
.
Eq. 2
where Φ’2 is the pure mode shape,
Φ2 is the polluted mode shape,
Φ1 is the mode to be swept out, and
Φ1TMΦ2/Φ1
TMΦ1 is the percent of the mode being swept out.
In order to use Equation 2, it was assumed that Mode 1 was the pure mode and
that mode 1 must be swept out of the higher modes. Mode 1 was assumed to be the pure
3.0 Vibration Testing 27
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
mode because it did resemble the expected mode shape, suggesting that there was little
pollution from higher modes. This assumption, that lower modes participate in higher
modes but not the reverse, was necessary to begin the process of sweeping modes. For
the purposes of this thesis, Mode 1 was assumed to be the pure shape and was then swept
out of mode 2. The limitation of this assumption is that higher modes do in fact
participate in lower modes, but it was impossible to sweep out a higher mode since the
shape was unknown. Without the pure shape of the higher mode it is not possible to
determine to what extent the higher mode is polluting the lower mode.
This concept of modal participation and modal sweeping is more easily explained
with a theoretical example of a beam with three modes: single bending, double bending
and triple bending as shown in Figure O on the following page. Assume that during
testing of this beam, Mode 2 was polluted with participation from Modes 1 and 3. The
shape of polluted Mode 2 is shown in Figure P on the following page.
3.0 Vibration Testing 28
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure O: Pure Modes 1, 2, and 3 Used in Theoretical Example
Source: Jacobsen (2011)
Figure P: Polluted Mode 2 Used in Theoretical Example
Source: Jacobsen (2011)
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0 50 100 150
Pure Modes
Mode 1
Mode 2
Mode 3
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 50 100 150
Mode 2 with 80% of Mode 1 & 40% of Mode 3
Mode 2 Pure
Mode 2 Polluted
Mode 2 Swept
3.0 Vibration Testing 29
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure P illustrates the many shapes that Mode 2 can appear as during testing for
a mode depending on the amount of modal pollution. The pure Mode 2 is shown as a
dashed line, the polluted Mode 2 is shown as a dotted line, and the Mode 2 after Mode 1
was swept from it is shown as a solid line.
In Figure P, it becomes clear that a modal participation from other modes can
obscure the pure shape of the mode. The pure Mode 2 is zero at the center of the beam,
but the polluted Mode 2 shows a positive value. The pure Mode 2 has a positive and
negative peak of equal value, whereas the polluted mode has a higher positive peak and a
shallow negative peak. Also illustrated in Figure P is Mode 2 after Mode 1 has been
swept from it. Again the inflection point of Mode 2 shifts due to the pollution of other
modes, in this case pollution from Mode 3. The Mode 2 swept shape is important to note
as it illustrates how the mode shapes of the buildings analyzed in this thesis may be
impure due to higher modal participation.
Mode 1 was pure enough for the purposes of this thesis and clearly participated in
Mode 2. In Figure N on the page 25, it is obvious that Mode 2 resembles Mode 1,
indicating a large amount of participation. When performing the Gram-Schmidt
sweeping method, the participation from Mode 1 in Mode 2 was calculated to be 82%.
So despite Mode 1 being impure, sweeping it from Mode 2 did improve the mode shape,
as seen in Figure Q following page.
Figure
Figure Q shows the two si
and though the shape does not resemble Mode 1 any
double bending with the inflection point located at the center
the two sides of the diaphragm moving opposite to each other.
Q could be attributed to several theories; different fixity at the ends of the diaphragm,
varying mass or stiffness along the diaphragm,
unaccounted flexibility in the diaphragm.
modal pollution and flexibility in the diaphragm,
With modal sweeping, e
Mode 1, sweeping an impure Mode 1 from Mode 2
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
0
Dis
pla
cem
en
t (f
t)
3.0 Vibration Testing
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure Q: Swept Mode 2 of Unretrofit Building
Source: Jacobsen (2011)
shows the two sides of the diaphragm moving opposite from each other
and though the shape does not resemble Mode 1 any longer, the shape is
with the inflection point located at the center. Also shown in Figure
the two sides of the diaphragm moving opposite to each other. The shape shown in Figure
several theories; different fixity at the ends of the diaphragm,
varying mass or stiffness along the diaphragm, modal pollution from higher mo
unaccounted flexibility in the diaphragm. Due to the complexity of the last two theories,
modal pollution and flexibility in the diaphragm, they will be discussed in more depth
With modal sweeping, even if it is assumed that Mode 2 is only pollut
Mode 1, sweeping an impure Mode 1 from Mode 2 will not yield perfect results
50 100 150 200 250
Length Along the Diaphragm (ft)
Unretrofit Building: Swept Mode 2
3.0 Vibration Testing 30
Retrofit Buildings
des of the diaphragm moving opposite from each other
not in pure
Also shown in Figure Q is
The shape shown in Figure
several theories; different fixity at the ends of the diaphragm,
from higher modes, or
last two theories,
will be discussed in more depth.
only polluted with
not yield perfect results, and the
South Side Experimental
North Side Experimental
3.0 Vibration Testing 31
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
clarity of Mode 2 is further obscured when higher modes, Mode 3 and Mode 4,
participate. As shown in Figure P for the theoretical beam example, sweeping Mode 1
from Mode 2 helps the shape but Mode 2 still looks distorted due to the higher modes. In
fact, the swept Mode 2 in Figure P resembles the mode shapes shown in Figure Q, with
the node shifted from center. It is highly probable that contribution from other modes
distorted the shape of Mode 2, shifting the inflection point from the center.
Aside from higher modal participation, an unaccounted flexibility in the
diaphragm could have yielded results similar to those tested. According to the shape
shown in Figure Q, the two sides of the diaphragm are moving in opposition to each
other. Since it is unlikely that the steel wide flanges are stretching and compressing to
allow for this shape, it is more probable that there is flexibility in the connections
between the roofing membrane and the actual steel frame beneath. The roof connection
is shown in Figure R on the following page.
3.0 Vibration Testing 32
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure R: Cross Section of Existing Roof
Source: Structural Plans of Existing Building (1959)
The load path through the roof includes multiple connections; the plywood is
nailed to a 3x10 wood joist that is then end-nailed to a 2x12 continuous wood member.
The 2x12 wood member is nailed to a 3x4 sill plate, and the 3x4 sill plate is bolted
through the top flange of the steel beam below. The indirect load path shown in Figure
R could allow for slippage in the connections, which could allow for the roof sides to
move opposite each other.
To try and mimic the slippage in the connections, a computational model was
created inserting springs between the plywood membrane and the steel frame, in an
attempt to model this flexibility. When the springs were stiff, the membrane moved with
the steel beams. When the springs were flexible, the membrane rotated on top of the
beams as though it were a rigid diaphragm. Neither of these behaviors matched the test
data, but they do not negate the possibility that unaccounted for flexibility in the
diaphragm could explain the results.
Figures S and T below
RB. The mode shapes were similar
were approximately 42% larger than the
763 µgs at the center of the diaphragm an
µgs at the center of the diaphragm
second mode were less significant and only varied 15%, with 1022
URB and 1210 µgs recorded on the
Figure S & T: Unswept Mode 1 (Left) and Mode 2 (right) of the Retrofit Building
The first mode shape for the
more symmetrical. The left and right quarter points were closer to the same acceleration,
matching the expected mode shape better than the
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 100
Dis
pla
cem
en
t (f
t)
Length Along Diaphragm (ft)
Retrofit Building: Unswept Mode 1
3.0 Vibration Testing
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
not negate the possibility that unaccounted for flexibility in the
diaphragm could explain the results.
below show the first and second mode shape, respectively, of the
. The mode shapes were similar to the URB but the accelerations recorded in the
were approximately 42% larger than the URB. The URB had a maximum acceleration of
gs at the center of the diaphragm and the RB had a maximum acceleration of 1342
at the center of the diaphragm. The differences between the accelerations of the
second mode were less significant and only varied 15%, with 1022 µgs recorded on the
gs recorded on the RB.
: Unswept Mode 1 (Left) and Mode 2 (right) of the Retrofit Building
Source: Jacobsen (2011)
The first mode shape for the RB appeared much the same as the URB
more symmetrical. The left and right quarter points were closer to the same acceleration,
matching the expected mode shape better than the URB. It should be noted that the lines
100 200
Length Along Diaphragm (ft)
Retrofit Building: Unswept Mode 1
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 100
Length Along Diaphragm (ft)
Retrofit Building: Unswept Mode 2
3.0 Vibration Testing 33
Retrofit Buildings
not negate the possibility that unaccounted for flexibility in the
st and second mode shape, respectively, of the
but the accelerations recorded in the RB
had a maximum acceleration of
acceleration of 1342
. The differences between the accelerations of the
gs recorded on the
: Unswept Mode 1 (Left) and Mode 2 (right) of the Retrofit Building
URB but was
more symmetrical. The left and right quarter points were closer to the same acceleration,
. It should be noted that the lines
200
Length Along Diaphragm (ft)
Retrofit Building: Unswept Mode 2
South Side Experimental
North Side Experimental
connecting these points are assumptions and that only the points wit
In addition, while testing the
to place the shaker in the same position
URB. The low platform is
building structure; it is thought that the platform was built to support future mechanical
equipment. Once all data was collected, the graph was flipped so that the mode shapes of
the two buildings could be more easily compar
Mode 2 for the RB
to be performed on the data. The swept Mode 2 is displayed below in Figure
Figure
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
0 50
Dis
pla
cem
en
t (f
t)
3.0 Vibration Testing
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
connecting these points are assumptions and that only the points with markers are known.
In addition, while testing the RB, a small wood platform on the roof made it impossible
to place the shaker in the same position on the RB as where the shaker was placed on
is built above the roof out of wood 2x4s and is not a part of the
building structure; it is thought that the platform was built to support future mechanical
Once all data was collected, the graph was flipped so that the mode shapes of
the two buildings could be more easily compared.
RB is polluted, similar to the URB, and the sweeping process had
to be performed on the data. The swept Mode 2 is displayed below in Figure
Figure U. Swept Mode 2 for the Retrofit Building
Source: Jacobsen (2011)
50 100 150 200 250
Length Along Diaphragm (ft)
Retrofit Building: Swept Mode 2
South Side Experimental
North Side Experimental
3.0 Vibration Testing 34
Retrofit Buildings
h markers are known.
on the roof made it impossible
re the shaker was placed on the
2x4s and is not a part of the
building structure; it is thought that the platform was built to support future mechanical
Once all data was collected, the graph was flipped so that the mode shapes of
, and the sweeping process had
to be performed on the data. The swept Mode 2 is displayed below in Figure U.
South Side Experimental
North Side Experimental
3.0 Vibration Testing 35
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure U shows that Mode 2 for the RB is distorted similarly to Mode 2 for the
URB shown in Figure Q on page 30. Just as the URB, the Mode 2 for the RB is shifted
so that the point at which the displacement is zero is not in the center of the diaphragm.
Also, the two sides of the diaphragm are moving opposite to each other. Modal pollution
and unaccounted flexibility in the diaphragm are possible reasons for this distortion.
While the mode shapes may have been distorted, they did succeed in confirming
that the retrofit did not affect the modal behavior for NS direction. The retrofit primarily
consisted of new braces added in the EW direction; braces in the EW direction should not
affect the building’s behavior in the NS direction as the two directions are orthogonal to
each other. The mode shapes for both buildings were nearly identical, with the highest
difference in recorded accelerations being 42% for Mode 1. This difference in
acceleration is explained by the smaller damping ratio in the retrofit building and does
not imply that the retrofit affected the NS modal behavior.
3.4 Damping Ratios
Both buildings showed clear natural frequencies in the NS direction but none in
the EW direction therefore, damping ratios were determined only for the NS modes.
Using the frequency-acceleration graphs seen in Figures D and H on pages 15 and 19
respectively, the damping ratios were found for both buildings using the half-power
bandwidth method equation shown on the next page,
3.0 Vibration Testing 36
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
n
ab
ω
ωωζ
2
−= . Eq. 3
where ξ is the damping ratio (unitless),
ωn is the resonant frequency (Hz), and
ωb and ωa are the frequencies at 2
1 the resonant amplitude (Hz).
Figure V below shows the frequency sweep for the URB in the NS direction and
the values used in the half-power bandwidth method outlined in Chopra (2007).
Figure V: Damping Ratios for the Unretrofit Building
Source: Jacobsen (2011)
The calculation for the damping ratio is easier to understand by examining Figure
V. The first mode had its peak at 3.35 Hz, the natural frequency for Mode 1, and the
3.0 Vibration Testing 37
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
amplitude for that peak is 25.4 µgs. The width of the peak indicates the amount of
damping in the mode; a steep peak on the frequency-acceleration graph indicates little
damping, and a broad peak indicated higher damping. The point at which the width of
the peak was evaluated was determined by dividing 25.4 µgs by √2, which equals
17.9 µgs for Mode 1. The two frequencies that correspond to this acceleration on either
side of the peak are 3.24 Hz and 3.5 Hz. Entering these frequencies, along with the
natural frequency, into Equation 3, the damping for Mode 1 is calculated to be 3.9%. The
same calculation was performed for Mode 2, which has a damping of 4.7%.
This process was repeated for the retrofit building, the results of which are seen
on the below in Figure W.
3.0 Vibration Testing 38
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure W: Damping Ratios for the Retrofit Building
Source: Jacobsen (2011)
The frequency-acceleration graph, as seen in Figure W, shows that the RB has
steeper peaks than the URB and therefore the damping is smaller. The damping for the
first mode is 2.2% and the second mode is 2.5% compared to the 3.9% and 4.7% for the
URB. Non-structural elements in a building can contribute to the damping of the
building and at the time of the testing, the RB was unoccupied and empty of any interior
partitions. Also, though the buildings are thought to be nearly identical, it is unknown
whether they were built at the same time. Variance in the construction methods and
3.0 Vibration Testing 39
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
materials could also account for the 50% difference in damping between the two
buildings.
3.5 Summary
Through FVT and AVT the natural frequencies, mode shapes, and damping ratios
of the two buildings in the NS direction was found.
The URB has a natural frequency of 3.35 Hz for the first mode and 6.31 Hz for the
second mode. The RB has a natural frequency of 2.66 Hz for the first mode and 5.63 Hz
for the second mode. The natural frequencies in the EW direction could not be detected in
either building.
The first two mode shapes in NS direction were detected in both buildings. The
first mode shape was simple bending of the diaphragm with maximum accelerations
detected in the center of the building. The second mode shape was expected to be double
bending, but the tested shape shows differential movement between the north and south
side of the building. It is possible that the slip between the plywood and the steel frame
resulted in a mode shape that appeared to be compressing and stretching. The plywood is
connected to wood joists that are then connected to the wide flanges so there are multiple
connections where that slip could occur. While no computational model showed this
behavior, it is still a possible explanation.
Another plausible explanation for Mode 2’s shape is higher model pollution. A
third peak on the FVT graph close to the second mode suggests the third mode is close to
the second, making it more likely to interfere with the second mode. As shown in Figure
3.0 Vibration Testing 40
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Q in the theoretical example, interference from a higher mode could distort the shape of
the second mode, explaining the shape found during testing.
The damping ratios for both buildings were also determined using the forced
vibration results in the NS direction. The unretrofit building had 3.9% damping in Mode
1 and 4.7% in Mode 2. The retrofit building had 2.2% damping in Mode 1 and 2.5%
damping in Mode 2.
From the testing it is apparent that the retrofit does not affect the NS modes. The
natural frequency shifts down between the URB and the RB suggesting an increase in
mass not stiffness. Since the braces should only stiffen the EW direction, this downward
shift aligns with the prediction. Also, the URB had no gravel on the roof while the RB
had several inches of gravel; this could account for the increase in mass. In addition to
the NS frequencies not increasing between buildings, the mode shapes of both buildings
match closely, indicating again that the new braces do not affect the NS direction.
4.0 Computational Modeling 41
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
4.0 COMPUTATIONAL MODELING
After the experimental data was gathered, the natural frequencies obtained from
the vibration testing were compared to the natural frequencies of computational models.
The two buildings were complex so the comparison process began with simple models
that could describe the building behavior. Subsequent comparisons used increasing model
complexity. By incrementally increasing the complexity of the model, the effects of the
different modeling assumptions on the building behavior were identified. During the
modeling process it became clear that the predicted mode shapes remained similar
through all iterations, simple bending of the diaphragm for Mode 1 and double bending
of the diaphragm for Mode 2, therefore the focus centered on comparing the natural
frequencies not the mode shapes. The modeling process was broken into two main
categories: hand calculations and computational modeling.
4.1 Hand Calculations
Freeman et al. (2002) state that the first mode of a Rigid-Wall/Flexible-Diaphragm
(RW/FD) building is governed by the deflection of the diaphragm in beam-like-bending
action. This perspective of viewing the diaphragm as a deep, thin beam warrants using
Timoshenko’s equations for the natural frequency of a simply supported beam (Rao,
2004). In Rao, (2004) several equations are considered for finding the natural frequency
of a beam depending on whether rotary inertia, shear deformation, or neither. The
equations that Timoshenko’s beam theory produces:
4.0 Computational Modeling 42
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
)1(2
2224
4422
l
rnl
nn
π
παω
+
= , Eq. 4
where ωn is the natural frequency (rad/s),
E is the modulus of elasticity (ksi),
I is the moment of inertia (in4),
ρA is the mass of the building per length of diaphragm(k*s2/ft/ ft),
n is the Nth mode,
l is the length of the diaphragm (in),
r2 is the radius of gyration squared (in), and
α2 is the modulus of elasticity multiplied by the moment of inertia
and divided by the seismic mass of the building per unit
length of diaphragm.
where only rotary inertia was considered,
)1(2
2224
4422
kG
E
l
rnl
nn
π
παω
+
= , Eq. 5
where k is the shear coefficient for the shape (unitless) and
G is the shear modulus.
where only shear deformation was considered, and
4
4422
l
nn
παω =
, Eq. 6
where neither rotary inertia nor shear deformation was considered.
4.0 Computational Modeling 43
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
The diaphragm was modeled in two ways, as a solid beam 5/8'' thick and 170.5'
deep; and as a beam with a hole in it the size of the sawtooth. Practical engineering
estimates considering the weight of the plywood, gravel, roofing, beams and half the
height of the columns and walls, were made when calculating the mass for the building.
The modulus of elasticity, E=1000 ksi, and shear modulus, G=417 ksi, of 5/8'' plywood
were used for the calculations. Mechanical properties of the plywood were obtained from
APA (1998) assuming S-1 grade level and dry conditions since the grade of the plywood
was not specified in the plans. It is also unknown how age has affected the strength of
the wood. All three equations were used for both beam models and compared to the
vibration results found in field as shown below in Table 1 below.
Table 1: Hand Calculation Results for NS Direction
Source: Jacobsen (2011)
The second column in Table 1 shows the predicted frequencies for the NS
direction assuming the diaphragm behaves like a flexible beam. The third column shows
the first natural frequency obtained from testing. It is important to note that the predicted
frequencies were compared to the frequencies of the URB found during testing. The
retrofit did not affect the NS modes, but the two buildings did have different masses due
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Forced Vibration Testing of Pre- and Post- Retrofit Buildings
to the gravel roofing material on the RB. The mass of the URB was used in the hand
calculations and so was compared to the NS frequencies of the URB.
Finally, the fourth column shows the percent difference between the frequencies
obtained analytically and the frequencies obtained through testing. The percent
difference found between the model frequency and the test frequency can be applied to
the RB. The percent difference between the frequencies increased when the diaphragm
was modeled as a beam with a hole in it, suggesting that the diaphragm acts as a solid
unit despite the sawtooth. Upon observing the percent difference of 31.6%, it was
concluded that the simple hand calculation was a fair predictor of the natural frequency
considering the simplicity of the calculation. The simple beam model indicates that more
than just the membrane contributes to the stiffness of the diaphragm; therefore, a more
complicated model was needed to replicate the modal behavior of the building.
While the diaphragm in the NS direction acted like a beam, both experimentally
and computationally, the diaphragm in the EW direction acted more like a rigid
diaphragm. This rigidity was expected considering the 0.63:1 aspect ratio of the
diaphragm was well below the code defined 3:1 ratio for flexible diaphragms. This
aspect ratio, coupled with the lack of stiff shearwalls in the EW direction, indicated that
the relative rigidity of the diaphragm was closer to the rigidity of the lateral system and
therefore the diaphragm did not behave like a flexible diaphragm. Consequently, the
natural frequency in the EW direction was calculated as though the diaphragm were rigid,
only considering the columns and out-of-plane stiffness of the two shearwalls.
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Forced Vibration Testing of Pre- and Post- Retrofit Buildings
The first computational model for the EW direction accounts for the out-of-plane
stiffness of the walls only. The second computational model added the stiffness of the
four moment frames. Then the third and final hand calculation includes the stiffness of
the new braces in order to predict the frequency of the retrofit building. The results for
these hand calculations are shown in Table 2 below.
Table 2: Hand Calculation Results for EW Direction
Source: Jacobsen (2011)
Though Table 2 does not show a comparison between tested frequencies and
analytical frequencies, the values do contribute to the mode shape prediction. The first
two models represent the URB and those models predict frequencies near 2.5 Hz. The
third model represents the RB with the braces and predicts a natural frequency of 5.61
Hz. This large shift in natural frequency suggests that once the braces are added, the
diaphragm no longer behaves as a rigid diaphragm. The significant increase in the LFRS
system stiffness indicates that the diaphragm is no longer relatively stiff and so bends
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Forced Vibration Testing of Pre- and Post- Retrofit Buildings
more like a beam, despite the aspect ratio. This change in modal behavior is also
observed in the computational modeling discussed later in this section.
4.2 Computational Modeling
The initial computer modeling consists of a simple membrane model of the
diaphragm. The membrane modeling is similar to the hand calculations for the beam
models and is an appropriate model to begin with based on the comparisons of the hand
calculations to the testing. The membrane has the dimensions of the diaphragm and is
meshed to allow flexibility between the pinned supports along the lines where the
shearwalls are located.
Various models with different membrane thicknesses were created in order to
observe how different modeling assumptions affect the modal response. It was important
to first determine whether the membrane model could have the gross section properties of
the plywood, or whether it had to be adjusted to account for the nonlinear behavior of a
wood diaphragm.
A flexible wood diaphragm cannot normally be viewed as one continuous
member. The diaphragm is composed of multiple plywood panels, nails, and chords, so
the deflection of a wood diaphragm is based on the slip and deflection of the multiple
components. The deflection of these multiple components leads to a higher deflection
than if the diaphragm is assumed to be a solid piece of plywood. Since the connections
of the plywood could not be modeled, the membrane was made thinner in order to match
the deflection from the diaphragm deflection equations. After analyzing the natural
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frequency associated with each deflection, it was deemed appropriate to use the thickness
associated with gross section properties. Due to the low force level used in testing, the
plywood diaphragm behaves more as a unit than as individual panels.
The responses of the models are compared to the unretrofit building and are
shown below in Table 3 below.
Table 3: Membrane Model Results for the NS Direction
Source: Jacobsen (2011)
None of the membrane models include vertical members or even beams, simply a
flat membrane with the properties of the plywood. The initial model’s membrane has a
reduced thickness so that the displacement of the diaphragm due to a uniform load
matches the IBC (2006) diaphragm displacement equation:
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Forced Vibration Testing of Pre- and Post- Retrofit Buildings
b
XLe
Gt
L
EAb
L C
n2
)(188.0
48
5 3 ∑ ∆+++=∆
νν , Eq. 7
where ν is the maximum shear due to design loads (plf),
L is the diaphragm length (ft),
E is the modulus of elasticity of the chord (psi),
A is the area of the chord cross section (in2),
b is the diaphragm width (ft),
Gt is the panel rigidity through the thickness (psi),
en is the nail or staple deformation (in),
∆ is the calculated deflection (in), and
Σ(∆CX) is the sum of individual chord-splice slip values on both
sides of the diaphragm, each multiplied by its distance to
the nearest support.
This calculation considers nail slip, chord slip, and chord bending in addition to
the shear deflection from the plywood. Chord and nail slip are considered to be zero due
to the small force of the shaker. The IBC equation allows for the greatest amount of slip
and therefore deflection, so the thickness of the membrane is small in order to decrease
the stiffness of the membrane and match the IBC deflection. The frequency of the first
membrane model, 1.088 Hz, is 67.5% different from the tested frequency of the unretrofit
building. This difference prompted the creation of models that were increasingly stiff.
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The IBC equation assumes the only chords are at the outermost edges of the
diaphragm, however, the buildings tested have chords at each frame line. Lawson (2007)
expanded on the IBC deflection equation and calls for added stiffness due to reduced
chord bending. The diaphragm deflection was recalculated per Lawson (2007) and the
thickness of the membrane was increased to match the smaller deflection. Though
increasing the thickness to match the deflection given in Lawson’s equation increases the
frequency to 1.518 Hz, the percent difference remains high at 54.7%. As a result, it was
determined that the membrane thickness should not be reduced at all.
The membrane was then modeled with the gross section properties of the
plywood. This increases the natural frequency significantly, up to 2.059 Hz, and the
percent difference drops to 38.5%, which is almost half of the original membrane model.
Though the percent difference is lower, more iterations were crafted to reduce the
difference further. It is not realistic to increase the thickness beyond the gross section
properties however, it is appropriate to alter the support conditions of the membrane. The
previous membrane models have pinned supports similar to those of a simply supported
beam, yet the diaphragm appears to be behaving more like a fixed-fixed beam. The
stiffness of the concrete shear walls is significant enough to assume that it restrains the
diaphragm, similar to a fixed condition. Therefore the final iteration of the membrane
models uses a fixed-fixed membrane with gross section properties of the plywood.
Changing the support conditions from pinned to fixed raises the natural frequency to
2.715 Hz and lowers the percent difference to 19%.
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Once the membrane models were completed, it was apparent that the modal
behavior of the diaphragm could not be modeled by the plywood membrane alone. Not
only are the membrane models too flexible, they are also limited in the mode shapes they
can produce. Though the model predicts simple bending in the NS direction, similar to
the test data, the model cannot predict any EW modes since the model is restrained in the
EW direction. The membrane model is ineffective at predicting the natural frequencies
with accuracy and does not yield much insight into the mode shapes of the building.
Despite this, the membrane model is useful for determining the correct effective
thickness of the membrane, which at the low level of shaking in this thesis, is the gross
section properties of the plywood. In common practice, wood diaphragms are modeled
with a thinner membrane in order to account for the additional flexibility in the
diaphragm due to slip between the plywood panels. It is important to prove that at the
low level shaking used in testing, the panels do not slip, and therefore the membrane in
the model does not need to be thinner than the gross section properties.
After the membrane model options were explored, a more complex model of the
diaphragm was created to further lower the percent difference between the tested and
predicted natural frequencies. This next stage of modeling introduces the steel beams,
girders, and eventually the sawtooth framing. The lateral-force-resisting system (LFRS)
is still excluded so only the members in the plane of the diaphragm affect the stiffness.
The goal of the diaphragm models is to explore how the steel framing, most specifically
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the sawtooth framing, affects the behavior of the building. The results of the diaphragm
models are shown in Table 4 below.
Table 4: Diaphragm Model With Sawtooth Results
Source: Jacobsen (2011)
The first diaphragm model includes the major wide flange beams, no joists, and a
hole in the membrane where the sawtooth is located. The east and west edges of the
diaphragm are fixed rather than pinned to simulate the fixedness of the diaphragm
connection to the shearwalls. The first model ignores any stiffness that the sawtooth
framing might contribute to the diaphragm and models a hole where the sawtooth is
located. The low frequency, 1.234 Hz, and high percent difference, 63.2%, suggests that
the sawtooth is not the equivalent of an opening in the diaphragm and does contribute
stiffness.
Based on these findings, a simplified version of the sawtooth is explored first.
The sawtooth is simplified by not modeling the slanted framing. Instead, the second
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model uses a flat membrane with a reduced thickness to simulate the stiffness of a slanted
membrane. On site, the plywood of the sawtooth is slanted, so one edge of the membrane
connects with the horizontal plywood of the diaphragm. This effect is simulated by
modeling the sawtooth as a horizontal membrane with only one edge connected. The
stiffness of the diaphragm does increase and the frequency shifts to 2.305 Hz, as shown
in Table 4 on the previous page, and the percent difference decreases to 31.2%.
This attempt to model the sawtooth appears viable, but the next step was to model
the sawtooth as it exists on site, with steel framing and slanted plywood. The sawtooth
framing consists of slanted wide flanges that connect to the frames below and small tube
columns that are also welded to the frame lines. Plywood spans between the wide flanges
along the slanted plane, and glass windows occupy the vertical plane. The steel framing
is included in the model as is the plywood membrane; the windows are excluded.
Modeling the sawtooth as it really is framed makes the diaphragm more flexible than the
flat membrane model, and the frequency reduces to 1.402 Hz. The percent difference
between this model and the tested frequencies increases to 58.2%.
The diaphragm model does not yield smaller percent differences than the
membrane model because of the large discontinuity in the diaphragm due to the sawtooth.
Though the membrane models have smaller percent differences, the building is not better
modeled with only a membrane. There are too many members ignored in the membrane
model for it to be anything other than a stepping stone to a more complete model. The
same is true for the diaphragm models. The diaphragm model is limited to predicting NS
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modes, since it is restrained in the EW direction; and the diaphragm does not predict the
natural frequencies with accuracy. It was initially assumed that since the mode shape of
the building in the NS direction is the diaphragm bending, the LFRS has little to do with
the mode shape, but after completing many diaphragm models it becomes clear that the
LFRS must affect the modal behavior of the diaphragm and should be included.
The final stage of modeling includes the vertical members, columns and shear
walls that support the diaphragm as shown in Figure X.
Figure X: Computer Model With Vertical Members Included
Source: Jacobsen (2011)
The vertical members provide the rigidity the diaphragm needs to accurately
represent the test results. Also by modeling the LFRS, the retrofit scheme can be
included in the model and compared to the test results to see whether the effect of the
retrofit can be predicted by the computational model. Since the walls extend from the
story below and have a thick waffle slab framing into them, the walls are modeled as
fixed at the base. The steel columns end at the waffle slab and the connection at their base
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is not stiff enough to suggest a fixed connection, therefore the columns are modeled as
pinned at the base. The comparison between the NS natural frequencies of the model to
the NS natural frequencies obtained during testing are shown in Table 5 below.
Table 5: 3-D Model Results for NS Direction
Source: Jacobsen (2011)
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The first 3-D model is of the RB and includes the shearwalls and the moment
frames. Among the diaphragm models, the model with the hole at the location of the
sawtooth is too flexible and poorly matches the tested natural frequencies. However, the
addition of the LFRS dramatically increases the stiffness of the building so the first
model includes the hole to determine whether the combination of the hole and LFRS
system matches the tested frequencies. This is a reasonable assumption, since many
engineers would discount any load transfer through the sawtooth, however, even with the
LFRS, the model is too flexible. The natural frequency of the 3-D model with the hole in
the diaphragm is 2.6 Hz, with a percent difference of 22.4%. This percent difference is
the largest shown in Table 5 and proves the sawtooth does not act as simply a hole in the
diaphragm according to the testing results.
At this stage in the modeling process it became important to look at the story
height and the level of the steel girders as they affected the moment frame stiffness; the
girders in the two directions are at two different levels as shown in Figure Y below.
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Figure Y: Section Showing Moment Frame Beams at Different Elevations
Source: Jacobsen (2011)
As seen in Figure Y, the girders in the EW direction frame on top of the NS
girders and so there are two top-of-steel elevations: 15' and 13.25'. Due to the complexity
of modeling the beams at the different levels and attaching the membrane to the two
levels, two models are created with the two different story heights.
The second model shown in Table 5 has a story height of 15' and also includes the
sawtooth framing which helps stiffen the diaphragm. By adding in the sawtooth framing,
the natural frequency of the building increases to 2.86 Hz and the percent difference
drops to 14.6%, the lowest of the models thus far. The third model shown in Table 5 is
similar to the second except that its story height is 13.25'. The frequency of the model
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with the lower story height is the closest to the tested frequency, with 3.23 Hz and a
percent difference of 3.6%.
The last model created for the unretrofit building, the fourth shown in Table 5 on
page 54, is identical to the previous model but without wood joists. As expected, the
frequency is barely affected by the change and the frequency only drops to 3.21 Hz.
The lower half of Table 5 illustrates the frequencies of the retrofit building
models. Most of the changes made for the unretrofit model are made for the retrofit
model, but the frequencies are lower due to the increased mass of the retrofit building.
Braces have been added to the building, but they do not affect the frequencies in the NS
direction.
The retrofit model started at a height of 15' and includes the sawtooth framing.
Because the hole in the diaphragm creates too much flexibility in the unretrofit model, it
is not included in the retrofit models. The model with the 15' roof height and all the
beams, matches the tested frequency most closely at 2.57 Hz and a percent difference of
2.3%. The small joists are excluded for a second model and this exclusion changes the
frequency marginally to 2.55 Hz and a percent difference of 3.0%. These models are
repeated with a lower roof height of 13.25', but the change in height stiffens the building
so that the frequencies become 2.84 Hz and 2.86 Hz, exceeding the tested frequency by
8.0% and 8.7%.
Once the LFRS was modeled, the natural frequencies improved noticeably. As
shown in Table 5, the highest percent difference among the 3-D models is 22.4%, much
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lower than the 58.4% difference in the membrane model. By including the LFRS the
computational model is able to confirm the results found in testing.
The natural frequencies found during testing suggest that the retrofit does not
affect the modal behavior of the building in the NS direction. This data is supported by
the computational models. The natural frequencies of the retrofit building decreased
during testing, due to the larger mass of the retrofit building. The RB has a thick layer of
gravel roofing material while the URB has a minimal amount. Had the braces affected the
NS direction, the frequency would have increased. This frequency shift is also observed
in the computational modeling and is shown in Table 5 on page 54. Since the retrofit is
primarily focused on the EW direction, the effect on the NS direction should be minimal,
and both testing and computational models support that result.
The first NS mode mapped during testing matches the mode from the
computational model well. Both the mode shape found during testing and the mode
shape from the computational model are single bending with the maximum displacement
in the center of the diaphragm. The only difference between the two is that the mode
shape from the computational model is symmetrical about the center of the diaphragm,
whereas the mode shape from the testing is lopsided as shown in Figure M on page 25.
This asymmetry about the center of the diaphragm is likely due to modal participation
from higher modes; an issue that is not present in the computational model. In addition to
their similar shape, both the testing and the computational model show no change in the
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Forced Vibration Testing of Pre- and Post- Retrofit Buildings
first NS mode due to the retrofit. Both the testing and the computational analysis suggest
the retrofit only affects the EW direction.
The second NS mode shape found in testing does not resemble the computational
model to the same degree as the first mode shape. The computational model shows the
second NS mode shape as the entire diaphragm in double bending; where the testing
shows a shape that is not double bending with two sides of the diaphragm acting in
opposition rather than in unison. While Mode 1 may be slightly polluted by modal
participation, Mode 2 is hardly recognizable due to the influence of other modes. Figure
P on page 28 shows how the mode shape can be obscured due to the participation of other
modes. Also, the closer modes are to each other on the frequency range, the more
participation is likely to occur. Figures D and H on pages 15 and 19 both show a peak
soon after the second mode, near 7 Hz, which may be the third mode. This potential third
mode is close enough to the second mode to suggest it is interfering with the test data.
Computational models were created to simulate unaccounted flexibility, another
reason for the obscured Mode 2. Springs were added to allow the plywood membrane to
move independently from the steel beams. When the springs became more flexible the
membrane began to behave like a rigid diaphragm, translating and rotating as a unit.
While the two sides of the diaphragm move in the same direction, even with the springs,
one side does not move as much as the other. This does not resemble the test data, and
the frequencies are disparate enough from the test data that the model results are not
included in Table 5. While unaccounted flexibility can exist in the building diaphragm, it
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Forced Vibration Testing of Pre- and Post- Retrofit Buildings
is not captured in the models. Another explanation for the difference between the tested
Mode 2 and the predicted Mode 2 is modal pollution from higher modes.
Through analysis of the 3-D models in the NS direction it is concluded that a
“good” model is a model that accurately replicates the real modal behavior of the
building, including major beams and girders, the sawtooth framing, and the LFRS. It is
not enough to model just the membrane or even just the roof framing; the entire system
has to be modeled before satisfactory results are achieved. However, it is apparent that
while the story height does have an effect, the difference is not so large between the two
heights that the models become inaccurate. It was also clear that a “good” model does not
require minute detailing; the wood joists have little effect on the modal behavior, so only
major elements need to be included in the model.
Another advantage of the 3-D models is that they allowed movement in the EW
direction and so frequencies can be obtained for that direction. Table 6 on the following
page illustrates how the changes to the 3-D model affect the natural frequency in the EW
direction.
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Table 6: 3-D Model Results for EW Direction.
Source: Jacobsen (2011)
Though there is no tested frequency for comparison, Table 6 does show the
change in the natural frequency due to the retrofit. The first model shown in Table 6, the
15' URB, is the most flexible with a frequency of 1.62 Hz. The lower roof height of
13.25' in the second model raises the natural frequency to 1.86 Hz. Again, the exclusion
of the wood joists lowers the frequency minimally, to 1.86 Hz. Adding the braces does
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increase the frequency to 2.51 Hz, but this number is deceptive as the additional mass of
the RB lowers the frequency, while the added stiffness of the braces raises it.
The next iteration of modeling was to lower the retrofit building height to 13.25'
and the frequency subsequently increases to 3.89 Hz. Both RB models have iterations
that exclude the joists. The frequency for the model with the 15' height is 2.48 Hz and the
model with the 13.25' height is 3.89 Hz.
The 3-D model is also the first to provide mode shapes in the EW direction since
the direction is no longer restrained. According to the computational model, the mode
shape in the EW direction for the unretrofit building is similar to that of a rigid
diaphragm as the entire roof displaces in the EW direction as a unit. Once the braces are
added, the edges of the diaphragm are held in place and the middle of the diaphragm
displaces in the EW direction so that the diaphragm resembles a deep beam.
Though there are no EW frequencies and mode shapes from the test data to
compare with the computational results, the computational results are instrumental in
determining whether the natural frequencies of the building are in the range of testing.
According to the computational model, the natural frequencies for the unretrofit building
range from 1.62 Hz to 1.90 Hz, which is in the range of the ambient vibration test (AVT).
Similarly the retrofit building frequencies range from 2.48 Hz to 3.89 Hz which is in the
range of the AVT and the forced vibration test (FVT). By using the mode shapes and
frequencies obtained from the computational model it is possible to calculate, through
modal analysis, what the recorded accelerations should have been due to the 30-lb shaker.
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There was initial concern that the reason the EW modes were undetected was that the
shaker did not produce large enough accelerations.
4.3 Modal Analysis
Through the modal analysis detailed below it is determined that had the natural
frequency been in the range of testing, it would have been detected.
The building is simplified into a nine degree-of-freedom (DOF) structure by
splitting the diaphragm into three sections, each with three DOFs. The nine DOF
structure are then excited by a 30-lb harmonic force and the accelerations due to that
excitation are calculated through modal analysis detailed in Chopra (2007).
Mü + Ku = p, Eq.8
where M is the mass matrix,
K is the stiffness matrix,
ü is the acceleration,
u is the displacement, and
p is the forcing function.
u = Φq , Eq. 9
where Φ is the mode shape and
q is the modal coordinate.
Equation 9 is substituted into equation 8 and then orthonormalized by multiplying
both sides of the equation by ΦT, shown in equation 10, which simplifies to equation 11
on the following page.
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ΦTΦ + ΦTΦq = ΦTp Eq. 10
I + Ω2q = ΦT
p, Eq. 11
where I is the identity matrix and
Ω is a matrix with the natural frequencies on the diagonal.
Due to the orthonormalization, Equation 8 is able to be uncoupled and become
equation 11. Equation 11 can separate into single-degree-of-freedom equations,
(equation 12.)
+ ωn2q = P0sin(ωt), Eq. 12
where P0sin(ωt) is the forcing function and
ωn is the natural frequency of the nth mode.
Once q is solved in equation 12, the displacement u can be calculated using
equation 9. The second derivative of u, shown in equation 13, yields the acceleration of
the diaphragm at the given point.
ü = uω2 Eq.
13
These calculations predict an acceleration of approximately 900 µgs for the first
mode in the NS direction for both buildings. These are comparable to the accelerations
found during testing which range from 760 µgs to 1300 µgs.
The estimates for the accelerations in the EW direction for the URB are 550 µgs
and the accelerations in the EW direction for the RB are 6 µgs.
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The predicted accelerations drop from 550 µgs at 1.86 Hz to 6 µgs at 3.89 Hz due
to the change in the EW mode shape during retrofit. According to the model, once the
braces are added, the LFRS becomes sufficiently stiff enough that the diaphragm behaves
like a flexible diaphragm between the brace lines. The mode shape, which originally was
defined as a rigid body translation in the EW direction, behaves like a beam, bending
between the brace lines. During the frequency sweeping, the accelerometers were placed
near the brace line, which according to the new mode shape, barely moves. Assuming
the computational model accurately predicts the mode shape of the building, this explains
why the EW direction is not detected for the retrofit building. Because the sawtooth
framing made the center of the roof inaccessible, the accelerometers were not placed in
the optimum spot to excite the EW mode of the RB. However, the modal analysis clearly
indicates that the first EW mode should have been identified for the unretrofit building if
it was in the tested range.
According to the computational models, the EW mode should be approximately
1.86 Hz for the URB and 3.89 Hz for the RB. The ambient sweeps include these
frequencies. Figure Z on the following page shows that there is no peak to indicate a
mode in the EW direction.
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Figure Z: Ambient Vibration of the Unretrofit Building in the EW Direction
Source: Jacobsen (2011)
As seen in Figure Z, the range from 1 Hz to 6 Hz is virtually flat. The only peaks
indicated in the EW ambient sweep are the peaks due to the mechanical units in the
building that oscillate at the higher frequencies. If there is an EW mode at the frequency
indicated by the computational models, it would have been recorded on the ambient
vibration test. Despite the small level of excitation, the AVT proved to be a reliable
indicator of what the natural frequency would be. In the NS direction, the AVT shows the
first mode which the FVT also confirms. The ambient vibration for the NS direction is
shown in Figure AA on the following page.
0.00E+00
5.00E-06
1.00E-05
1.50E-05
2.00E-05
2.50E-05
3.00E-05
3.50E-05
4.00E-05
4.50E-05
5.00E-05
0 2 4 6 8 10 12 14 16
Acc
eler
atio
n (
µg)
Frequency (Hz)
Ambient Test of the Unretrofit Building in the EW Direction
4.0 Computational Modeling 67
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure AA: Ambient Vibration of the Unretrofit Building in the NS Direction
Source: Jacobsen (2011)
In Figure AA, the peaks at 3.35 Hz and 6.18 Hz can be discerned in the ambient
vibration, clearly indicating that the AVT is sensitive enough to detect a mode if it is
within the range of testing. Similarly, the EW ambient vibration test for the retrofit
building did not detect an EW mode as illustrated in Figure BB on the following page.
0.00E+00
5.00E-06
1.00E-05
1.50E-05
2.00E-05
2.50E-05
3.00E-05
3.50E-05
0 2 4 6 8 10 12 14 16
Acc
eler
atio
n (
µg)
Frequency (Hz)
Ambient Vibration of the Unretrofit Building in the NS
Direction
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Figure BB: Ambient Vibration of the Retrofit Building in the EW Direction
Source: Jacobsen (2011)
The predicted natural frequency for the RB is 3.89 Hz. As is clear in Figure BB,
there is no peak near 4 Hz. The peaks indicated appear at the higher frequencies most
likely due to the mechanical equipment located on the roof. That the AVT could detect
the natural frequencies in the NS direction, shown in Figure CC on the following page,
suggests that if the EW natural frequencies are within the range of the test, a peak should
have appeared in Figure BB.
0.00E+00
5.00E-06
1.00E-05
1.50E-05
2.00E-05
2.50E-05
3.00E-05
3.50E-05
2 4 6 8 10 12 14
Acc
eler
atio
n (
µgs)
Frequency (Hz)
Ambient Vibration of the Retrofit Bulding in the EW
Direction
4.0 Computational Modeling 69
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Figure CC: Ambient Vibration of the Retrofit Building in the NS Direction
Source: Jacobsen (2011)
In Figure CC the peaks at the natural frequencies, 2.63 Hz and 5.58 Hz, are
clearly discernible. These clear peaks are in contrast to Figure BB where no peaks appear
at the predicted frequency range. The ambient vibration is capable of detecting the
frequencies in the NS direction and it should have detected the frequencies in the EW
direction. From the previous graphs it is apparent that had the natural frequencies fallen
within the tested range, the AVT should have detected them. Failing that, the FVT would
have detected them.
Previous testing using the same equipment, (McDaniel and Archer 2009) has
always managed to locate the first mode of the building in each direction. As of yet, the
testing of these buildings in the EW direction appears to be the only case where the first
mode was not detected with this method of testing. That the NS modes were easily
0.00E+00
2.00E-06
4.00E-06
6.00E-06
8.00E-06
1.00E-05
1.20E-05
1.40E-05
1.60E-05
1.80E-05
2 4 6 8 10 12 14
Acc
eler
atio
n (
µg)
Frequency (Hz)
Ambient Vibration of the Retrofit Bulding in the NS
Direction
4.0 Computational Modeling 70
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
detected in both ambient and forced vibration testing and that testing on other buildings
yielded results, suggests that the method isn’t flawed but rather the predicted frequencies
are inaccurate. Previous success with the testing method suggests that the natural
frequencies of the EW modes are either higher than the tested range, above 18 Hz, or
lower than the tested range, below 1.5 Hz. Again, though the computational model may
have been unsuccessful at predicting frequencies in the EW direction, the NS and EW
modes are uncoupled so the computer model still successfully predicts the modes for NS
direction.
4.4 Summary
The modeling process demonstrated that simplified models, such as hand
calculations and the membrane models, did not accurately predict either building’s modal
behavior. The smallest percent difference between the hand calculations and the tested
frequencies is 31.6%; the smallest percent difference between the membrane models and
the tested frequencies is 19%. Though the simple models cannot accurately predict the
natural frequencies they are instrumental in understanding the basic behavior of the
building diaphragm.
The membrane model helped conclude that the gross section properties of the
plywood should be used, rather than a thinner, more flexible membrane. The more
complicated 2-D diaphragm models are better predictors of the building behavior but
ultimately still underestimate the natural frequency by 31.2%. The large difference
4.0 Computational Modeling 71
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
between the diaphragm model frequencies and the tested frequencies prompted the
creation of models that include the LFRS.
The final iteration of models includes the columns, walls, and braces in the EW
direction and is able to accurately predict the building behavior. By including the LFRS
the model predicts a natural frequency within 2%-3%. Through the modeling process it
was determined that all the main members: beams, girders, columns, shearwalls, and
braces, must be included in the model in order to capture the building’s behavior. The
girders in the direction of testing stiffen the diaphragm by decreasing the span of the
diaphragm and the girders in the transverse direction stiffen the diaphragm by effectively
increasing its section modulus. The LFRS must be included as they stiffen the diaphragm
by acting as springs throughout the diaphragm and restraining its movement. It is also
apparent that the model must include the sawtooth framing in order to represent the
building; the sawtooth does provide load transfer and therefore cannot be modeled as a
hole in the diaphragm. However, small members, such as joists, can be excluded from
the model without impacting the modal behavior.
The final model accurately predicts the NS natural frequencies, but it does not
accurately predict EW frequencies. Though the predicted EW frequencies are in the
range that was tested, no EW modes show up on the FVT graphs. Through modal
analysis it was assessed that if the model is correct in the EW direction, the accelerations
at the first two EW modes should have been detected. That the modes were not detected
indicates that the model does not accurately capture the EW direction of the building.
4.0 Computational Modeling 72
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
This does not impact the model’s ability to predict the NS direction, however, since the
two directions are uncoupled.
5.0 Building Responses 73
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
5.0 BUILDING RESPONSE
The goal of creating the different models was to determine what level of detail
was necessary to capture the behavior of the buildings. But in order to draw any
conclusions, the criteria to evaluate the models needed to be established. One criterion
was the natural frequencies of the models, and the other was the models’ response to the
two response spectra. One of the response spectra was the LA ground motion developed
by the SAC Joint Venture (FEMA 355C) with a 475-year return period. The other
response spectrum was the design spectrum published in the ASCE-7. Both spectra are
shown below in Figure DD.
Figure DD: Response Spectra Used in Comparison of Models
Source: Jacobsen (2011)
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1.400
1.600
1.800
2.000
0 0.5 1 1.5 2 2.5
Acc
eler
atio
ns
(g)
Period (Sec)
Response Spectrums
SAC
ASCE 7-05
5.0 Building Responses 74
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
It is beneficial to use spectra that are different from one another to ensure that the
effects of changes to the model are universal regardless of the spectrum used. According
to the ASCE-7 spectrum, a wide range of building frequencies have the same
acceleration. The SAC spectrum is more sensitive to changes in building frequencies,
making it a preferable spectrum for comparison. The two spectra have not been scaled to
each other, so no comparison can be made between the two specifically, only between the
different models.
Both response spectra are based on the experimental damping ratio obtained in
testing, 3%. Once all the criteria are recorded, the displacement at the center of the
diaphragm, the total base shear, and the shear in the shearwalls, the models can be
compared to each other and a conclusion can be drawn regarding what changes
significantly affect the building responses.
Models that were thought to be a good approximation of the building, based on
their ability to predict the natural frequency of the buildings, were chosen for this
comparison. The baseline model is considered to be the model that best matches the
experimental frequency. The best matched model is the 13.25' tall building with the
sawtooth, all roof framing members, and in the case of the RB diagonal braces. These
models are the baseline that the other models are compared against. The responses of the
models are shown in Table 7 below.
5.0 Building Responses 75
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
Table 7: Comparison of Responses for Different Unretrofit Building Models
Source: Jacobsen (2011)
Table 7 shows several model variations: the base model, Model 1, which models
all the beams, the sawtooth, and the LFRS; Model 2, which has a story height of 15'
rather than 13.25'; Model 3, which models only the major beams; Model 4, which models
the sawtooth as a hole rather than a slanted diaphragm; and Model 5, which does not
model the LFRS. Though more models were generated, these three are thought to be
most similar to those a practicing engineer might create. Modeling the building with a
15' story height, Model 2, produces a more significant change, with the diaphragm
deflection increasing by 18% and 26.4%. The model with only major beams included,
Model 3, differs only slightly from the base model, with the highest percent difference
being a 2% increase in shear in the walls. The model that produces the highest percent
5.0 Building Responses 76
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
difference is Model 4 with no sawtooth framing, only a hole in the roof center. Model 4
has a diaphragm deflection increase of 36% and 51.2%, and a base shear decrease of
11.7% and 2.7%.
One 2-D model is included, to prove that a 2-D model can not accurately predict
any response criteria, not just natural frequencies. It is apparent in Table 7 that the 2-D
model is the most disparate from the base model with an increase in diaphragm deflection
of 231.9% and 373.8%, and a base shear increase of 45.9% and 83%.
The same comparison is performed on the retrofit building and the results of the
response spectra on the models are shown below in Table 8.
Table 8: Comparison of Retrofit Building Responses
Source: Jacobsen (2011)
Table 8 is similar to Table 7 except no model was created of the RB with a hole in
the diaphragm. All models include the retrofit with the sawtooth framing and slanted
membrane. The two 3-D variations, Models 2 and 3, have some effect but remain close
5.0 Building Responses 77
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
to the baseline model. The model with only major beams, Model 3, barely affects the
deflection, by less than 1%, but does change the base shear by almost 10%. The model
with the 15' story height, Model 2, has a greater effect with almost a 20% increase in
deflection and approximately 12% decrease in base shear. The 2-D model, Model 4, is
significantly different from the baseline model as expected since it has no vertical
members. The base shear changes by 9% and 27.1%, and the deflection of the diaphragm
increases dramatically, 184.3% and 279%.
Ultimately, the largest effect due to the changes is the diaphragm deflection, with
the change in deflection ranging as high as 51% in the 3-D models and 374% in the 2-D
diaphragm models. This dramatic change in diaphragm deflection indicates a wide range
in responses depending on the assumptions that the engineer makes. However, if the
diaphragm deflection is still below the allowable limit, the engineer may not be
concerned regarding the overestimation of the deflection. Though the IBC (2006) limits
the span-to-width ratio of the diaphragm, the actual deflection limit of the diaphragm is at
the discretion of the engineer. Engineers must consider the effects of the diaphragm
deflection on the walls beneath the diaphragm and decide whether the deflection is
acceptable (Bryer et al. 2007). In the case of the 3-D models, the highest diaphragm
deflection is 3.1'', resulting in a story drift of 1.7%. This is an acceptable drift, so despite
the changes had on the diaphragm deflection, the results could still be considered
acceptable. Base shear and natural frequency change but to a lesser degree than the
diaphragm deflection.
5.0 Building Responses 78
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
The two response spectra analyses were done to the 2-D diaphragm model to
illustrate the importance the LFRS. Table 7 on page 75 and Table 8 on page 76 how
significantly the diaphragm deflection is affected by the interior moment frames.
Without the interior moment frames, the deflection increases by 231% and 373%.
Through this comparison it is apparent that the building cannot be accurately modeled
with a diaphragm alone, even if the only response quantity desired is diaphragm
deflection. The diaphragm and the vertical system cannot be evaluated separately since
the stiffness of the diaphragm partially depends on the system below it.
The response spectra analyses confirm that in order to accurately predict the
building response, the computational model must include the sawtooth framing, LFRS,
and all major beams, columns, and walls. As seen in the frequency comparison, the
exclusion of minor joists does little to the building response. However, contrary to the
frequency comparison, the spectra analyses concluded that a 10% change in height
changed the building response by a significant amount, anywhere from 10% to 24%. The
spectra analyses suggest that the height assumption can have a significant effect on the
building responses and therefore a proper model should consider the correct effective
height of the members.
6.0 Conclusions 79
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
6.0 CONCLUSIONS
The first two natural frequencies for both the buildings were found through forced
vibration testing (FVT). The first two natural frequencies for the unretrofit building
(URB) are 3.35 Hz and 6.31 Hz. The first two natural frequencies for the retrofit building
(RB) are 2.66 Hz and 5.63 Hz. Though the NS modes were easily detected in both
buildings, the EW modes were undetected and modal analysis later confirmed that,
despite computational model predictions, the EW natural frequencies were outside the
range of testing, 2 Hz to 10 Hz.
Without the EW natural frequencies, no mode shape testing could be done, so
only the NS modes were mapped. The first NS mode was simple bending of the
diaphragm with the maximum acceleration recorded in the center of the diaphragm. The
second mode shape looked similar to the first mode, indicating a large amount of modal
pollution from the first mode. However, even after the first mode was swept out of the
second mode, Mode 2 is not pure double bending and shows the two sides of the
diaphragm moving opposite to each other. One possible explanation for the shifted
double bending is varying stiffness or mass along the length of the diaphragm. The
stiffness of the diaphragm could vary over the length of the diaphragm due to the
sawtooth or inconsistencies in the construction. The sawtooth framing is significantly less
stiff than the flat diaphragm framing and that change in stiffness could affect Mode 2.;
Lumped mass at one end could cause the shift in Mode 2 and there were mechanical units
for both the RB and URB located at the west end of the building.
6.0 Conclusions 80
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
While the shifted mode shape could be explained by varying stiffness or mass, an
explanation for the two sides of the diaphragm moving opposite to each other could be
the plywood diaphragm flexing on top of the steel frame. The plywood is nailed to wood
joists that sit on top of the steel wide flanges, so there is a chance that the plywood moves
on top of the frame without engaging the wide flanges. Though this flexibility in the
connections is contrary to the assumption made during modeling, that the plywood
membrane acts as unit, the possibility could not be ruled out during testing.
Another likely explanation for the distortion of Mode 2 is modal pollution from
higher modes interfering with Mode 2. A third peak on the FVT graph close to the second
natural frequency suggests that Mode 3 is close to Mode 2 and participation from Mode 3
may be polluting the results for Mode 2.
In comparing the results of computational models to the test results of two stiff-
wall/flexible-diaphragm buildings it became clear that while simple hand calculations did
a fair job of approximating the natural frequency of the building, a more complex
computational model is required to accurately capture the building’s modal behavior.
The initial computational models garnered improved results from the hand calculations,
with more accurate natural frequencies and mode shape predictions in the NS direction.
Among the multiple membrane and 2-D diaphragm models the lowest percent difference
from the tested natural frequencies was 19%. Though the early models can predict mode
shapes in the NS direction, a drawback of models that excludes the vertical systems is
their inability to predict modal behavior in the EW direction. Due to the inaccuracy of
6.0 Conclusions 81
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
the frequency predictions and the limit of modal behavior in the EW direction, a more
complex model is necessary.
The models that are able to accurately predict the modal behavior in the NS
direction include the sawtooth framing, moment frames, shearwalls, and brace frames in
the case of the retrofit building. The sawtooth framing must be modeled to represent the
load path through the sawtooth. If the sawtooth framing is not well connected to the
diaphragm it may be more accurate to model a hole in the diaphragm, however in the
case of these buildings, there is a clear load path through the sawtooth and it is inaccurate
to exclude the sawtooth from the model. The girders are necessary as they help to
provide stiffness to the diaphragm. The diaphragm spans between the girders and so by
including the girders parallel to the direction of loading, the span of the diaphragm is
reduced and the diaphragm is stiffened. The girders perpendicular to the direction of
loading act as part of the diaphragm and increase its section modulus, therefore also
increasing its stiffness. Finally, the LFRS must also be included in the model as the
LFRS acts as springs between the diaphragm and the ground, providing a semi-rigid
connection. The deflection of the diaphragm cannot be accurately predicted without
consideration of the LFRS.
The model including the sawtooth framing, girders, and LFRS is able to predict
natural frequencies within 3% of the experimental results, the closest match of any of the
computational models, confirming the importance of including all of these components.
The model also predicts the first mode shape in the NS direction to be simple bending of
6.0 Conclusions 82
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
the diaphragm, similar to what the testing revealed. Though the model is successful at
predicting the NS direction, the EW predictions do not match the test results. Neither the
AVT nor the FVT tests detected any natural frequencies in the range the computational
model predicts, suggesting that the computational model does not accurately capture the
EW behavior of the buildings.
Though the EW natural frequencies were out of range of the testing, either lower
than 1.5 Hz (T=0.67 sec) or higher than 10 Hz (T=0.1 sec), effects of the retrofit can be
discerned from the other results. The retrofit scheme consists of braces in the EW
direction so the NS direction should not be affected. Through testing and computational
modeling, this prediction is confirmed as the natural frequencies of the retrofit building
shift down due to the increase in mass of the retrofit building, rather than up due to the
stiffening of the building. The mode shape in the NS direction is also unaffected by the
retrofit in both the testing and the computational modeling.
The experimental method works well for predicting the modal behavior of the
building. The buildings tested in this thesis are larger than any others tested with this
method, and all previously tested buildings consisted of rigid diaphragms rather than
flexible diaphragm. It was originally thought that the 30-lb shaker would be unable to
excite the entire building and that only the diaphragm membrane would be excited. Clear
results were acquired for the NS direction that were confirmed with the computational
model, verifying that FVT with the small shaker is able to predict the natural frequencies
of the building. The inability to predict the natural frequency in the EW direction is not a
6.0 Conclusions 83
Forced Vibration Testing of Pre- and Post- Retrofit Buildings
limitation of the testing equipment, rather it is thought that the EW natural frequencies lie
outside the range of interest, 2 Hz to 10 Hz, or in some cases that the equipment was not
ideally placed to excite the building. Overall, the mode shape testing was successful,
particularly at predicting Mode 1, but participation from higher modes made it difficult to
discern Mode 2. This amount of modal pollution is likely due to the flexible diaphragm
as previous testing on rigid diaphragms has been able to determine the first two and even
three modes of the buildings.
For future testing on flexible diaphragms it is recommended that modal analysis
of the building be completed before testing begins. Modal analysis can help predict
whether accelerations can be detected and also help predict the optimum locations to
excite the building. Better understanding of the predicted modes shapes could ultimately
lead to more conclusive testing, whether to prove or disprove the predicted shapes.
7.0 Works Referenced List 84
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7.0 WORKS REFERENCED LIST
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